LEARN · EXPLAIN · REVISE

Disprove the claim that n²+n+t is prime for every nonnegative integer n.

Read the idea, work independently, then explain what changed.

高二選擇性必修第二册(A版).pdf · 4.4 · PDF 49 / printed page 44

TOPIC 01

Mathematical induction

Build proofs with a verified base, an explicit induction hypothesis and a valid step.

Build proofs with a verified base, an explicit induction hypothesis and a valid step.
Justify the method and check the conditions in a new situation.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Standard#Your turn

t=12

Induction proves a statement only for the specified integer domain starting at the base case.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Find one valid counterexample.

Hint 2

Try n=t.

Worked solution

The requested relation or conclusion is shown below.

n=12:n^2+n+t=12·14

Checks and common pitfalls: Both factors exceed one, so the expression is composite.

Reasoning checklist · self / teacher assessment

Think first. Reveal a hint when the class is ready.

Teacher preparation and assessment

Build proofs with a verified base, an explicit induction hypothesis and a valid step.
Which condition is essential in mathematical induction?
Can a correct induction step rescue a false or missing base case?

Defining relation: Build proofs with a verified base, an explicit induction hypothesis and a valid step.
$P(n_0)\land[P(k)\Rightarrow P(k+1)]$
Conditions: Induction proves a statement only for the specified integer domain starting at the base case.